data_viz

options(encoding = 'UTF-8')
#Loading all the necessary packages
if (!require("CASdatasets")) install.packages("CASdatasets", repos = "http://cas.uqam.ca/pub/R/", type="source")
if (!require("caret")) install.packages("caret")
if (!require("ggplot2")) install.packages("ggplot2")
if (!require("mgcv")) install.packages("mgcv")
if (!require("plyr")) install.packages("plyr")
if (!require("gridExtra")) install.packages("gridExtra")
if (!require("visreg")) install.packages("visreg")
if (!require("MASS")) install.packages("MASS")
if (!require("plotrix")) install.packages("plotrix")
if (!require("rgeos")) install.packages("rgeos", type="source")
if (!require("rgdal")) install.packages("rgdal", type="source")
if (!require("xtable")) install.packages("xtable")
if (!require("formatR")) install.packages("formatR")
if (!require("maptools")) install.packages("maptools")

require("CASdatasets")
require("ggplot2")
require("mgcv")
require("caret")
require("gridExtra")
require("plyr")
require("visreg")
require("MASS")
require("plotrix")
require("rgdal")
require("rgeos")
require("xtable")
require("formatR")
require("maptools")

Introduction

At the beginning of every data analysis some data manipulations and visualizations are required. The core for this is the tidyverse universe, which is an opinionated collection of R packages designed for data science. All packages share an underlying design philosophy, grammar, and data structures.

https://www.tidyverse.org/

Learn the tidyverse: See how the tidyverse makes data science faster, easier and more fun with “R for Data Science”. Read it online (https://r4ds.had.co.nz/), buy the book or try another resource from the community.

For visualizations, tidyverse contains ggplot2, see https://ggplot2.tidyverse.org/, and the corresponding book at https://github.com/hadley/ggplot2-book.

Load data

## 1) If 'CASdatasets' package can be loaded, then as follows:
data("freMTPLfreq")
freMTPLfreq = subset(freMTPLfreq, Exposure <= 1 & Exposure >= 0 & CarAge <= 
    25)
# Subsample of the whole dataset
set.seed(85)
folds = createDataPartition(freMTPLfreq$ClaimNb, 0.5)
df_freqMTPLfreq = freMTPLfreq[folds[[1]], ]
# save(df_freqMTPLfreq, file='../dataset.RData')

## 3) If not, then download the file 'freMTPLfreq.RData' from the GitHub
## repository and run the following: data_path <-
## 'C:/Users/juerg/AktuarDataScience/Varia/2019_SSA/'
## load(file=paste(data_path,'df_freMTPLfreq.RData',sep=''))
dataset <- df_freqMTPLfreq

A good idea is to check whether the dataset has been loaded correctly. To do this, the following tools can be used:

• head allows to visualize the first 6 lines of the dataset.

head(dataset)

##   PolicyID ClaimNb Exposure Power CarAge DriverAge
## 1        1       0     0.09     g      0        46
## 2        2       0     0.84     g      0        46
## 4        4       0     0.45     f      2        38
## 5        5       0     0.15     g      0        41
## 7        7       0     0.81     d      1        27
## 9        9       0     0.76     d      9        23
##                                Brand     Gas Region Density
## 1 Japanese (except Nissan) or Korean  Diesel    R72      76
## 2 Japanese (except Nissan) or Korean  Diesel    R72      76
## 4 Japanese (except Nissan) or Korean Regular    R31    3003
## 5 Japanese (except Nissan) or Korean  Diesel    R52      60
## 7 Japanese (except Nissan) or Korean Regular    R72     695
## 9                               Fiat Regular    R31    7887

• str allows to see the format of the different variables. We will typically distinguish numerical variables (real numbers or integers) and factors (categorical data).

str(dataset)

## 'data.frame':    205432 obs. of  10 variables:
##  $ PolicyID : int  1 2 4 5 7 9 10 14 17 18 ...
##  $ ClaimNb  : int  0 0 0 0 0 0 0 0 0 0 ...
##  $ Exposure : num  0.09 0.84 0.45 0.15 0.81 0.76 0.34 0.19 0.8 0.07 ...
##  $ Power    : Factor w/ 12 levels "d","e","f","g",..: 4 4 3 4 1 1 6 2 2 2 ...
##  $ CarAge   : int  0 0 2 0 1 9 0 0 0 0 ...
##  $ DriverAge: int  46 46 38 41 27 23 44 33 69 69 ...
##  $ Brand    : Factor w/ 7 levels "Fiat","Japanese (except Nissan) or Korean",..: 2 2 2 2 2 1 2 2 2 2 ...
##  $ Gas      : Factor w/ 2 levels "Diesel","Regular": 1 1 2 1 2 2 2 2 2 2 ...
##  $ Region   : Factor w/ 10 levels "R11","R23","R24",..: 9 9 5 6 9 5 1 1 1 1 ...
##  $ Density  : int  76 76 3003 60 695 7887 27000 1746 1376 1376 ...

• summary allows to compute for each variable some summary statistics.

summary(dataset)

##     PolicyID         ClaimNb           Exposure            Power      
##  Min.   :     1   Min.   :0.00000   Min.   :0.002732   f      :47709  
##  1st Qu.:102981   1st Qu.:0.00000   1st Qu.:0.200000   g      :45390  
##  Median :206768   Median :0.00000   Median :0.530000   e      :38329  
##  Mean   :206581   Mean   :0.03914   Mean   :0.559871   d      :33720  
##  3rd Qu.:309942   3rd Qu.:0.00000   3rd Qu.:1.000000   h      :13346  
##  Max.   :413168   Max.   :4.00000   Max.   :1.000000   j      : 8962  
##                                                        (Other):17976  
##      CarAge         DriverAge    
##  Min.   : 0.000   Min.   :18.00  
##  1st Qu.: 3.000   1st Qu.:34.00  
##  Median : 7.000   Median :44.00  
##  Mean   : 7.417   Mean   :45.29  
##  3rd Qu.:12.000   3rd Qu.:54.00  
##  Max.   :25.000   Max.   :99.00  
##                                  
##                                 Brand             Gas        
##  Fiat                              :  8341   Diesel :102771  
##  Japanese (except Nissan) or Korean: 39413   Regular:102661  
##  Mercedes, Chrysler or BMW         :  9651                   
##  Opel, General Motors or Ford      : 18638                   
##  other                             :  4947                   
##  Renault, Nissan or Citroen        :108267                   
##  Volkswagen, Audi, Skoda or Seat   : 16175                   
##      Region         Density     
##  R24    :79767   Min.   :    2  
##  R11    :34778   1st Qu.:   67  
##  R53    :21032   Median :  287  
##  R52    :19321   Mean   : 1982  
##  R72    :15506   3rd Qu.: 1408  
##  R31    :13576   Max.   :27000  
##  (Other):21452

If one needs some help on a function, typing a question mark and the name of the function in the console opens the help file of the function. For instance,

?head

Descriptive Analysis of the portfolio

We will now have a descriptive analysis of the portfolio. The different variables available are PolicyID, ClaimNb, Exposure, Power, CarAge, DriverAge, Brand, Gas, Region, Density.

PolicyID

The variable PolicyID related to a unique identifier of the policy. We can check that every policy appears only once in the dataset

length(unique(dataset$PolicyID)) == nrow(dataset)

## [1] TRUE

Exposure in month

The Exposure reveals the fraction of the year during which the policyholder is in the portfolio. We can compute the total exposure by summing the policyholders’ exposures. Here we find 115015.5 years.

We can show the number of months of exposure on a table.

table(cut(dataset$Exposure, breaks = seq(from = 0, to = 1, by = 1/12), labels = 1:12))

## 
##     1     2     3     4     5     6     7     8     9    10    11    12 
## 31393 14729 16610 12040  9793 14680  9447  7248 10714  6838  6066 65874

Using the function prop.table, it is possible to represent this information in relative terms show the number of months of exposure on a table.

round(prop.table(table(cut(dataset$Exposure, breaks = seq(from = 0, to = 1, 
    by = 1/12), labels = 1:12))), 4)

## 
##      1      2      3      4      5      6      7      8      9     10 
## 0.1528 0.0717 0.0809 0.0586 0.0477 0.0715 0.0460 0.0353 0.0522 0.0333 
##     11     12 
## 0.0295 0.3207

Alternatively, we can use a barplot !

Exposure.summary = cut(dataset$Exposure, breaks = seq(from = 0, to = 1,by = 1/12))
levels(Exposure.summary) = 1:12
ggplot()+geom_bar(aes(x=Exposure.summary)) + xlab("Number of months") + ggtitle("Exposure in months")

Number of claim : ClaimNb

ggplot(dataset, aes(x=ClaimNb))+geom_bar()+
geom_text(stat='count', aes(label=..count..), vjust=-1)+ylim(c(0,210000))+
  ylab("")+ xlab("Number of Claims")+  ggtitle("Proportion of policies by number of claims")

We can compute the average claim frequency in this portfolio, taking into account the different exposures.

sum(dataset$ClaimNb)/sum(dataset$Exposure)

Here, we obtain 0.0699.

Let us now look at the other variables.

Power

The variable Power is a categorized variable, related to the power of the car. The levels of the variable are ordered categorically. We can see the different levels of a factor by using the function level in R:

levels(dataset$Power)

##  [1] "d" "e" "f" "g" "h" "i" "j" "k" "l" "m" "n" "o"

We can see the number of observations in each level of the variable, by using the function table.

table(dataset$Power)

## 
##     d     e     f     g     h     i     j     k     l     m     n     o 
## 33720 38329 47709 45390 13346  8793  8962  4659  2255   905   624   740

Remember however, that in insurance, exposures may differ from one policyholder to another. Hence, the table above, does NOT measure the exposure in each level of the variable Power. We can use the function ddply to give us the exposure in each level of the variable.

require(plyr)
Power.summary = ddply(dataset, .(Power), summarize, totalExposure = sum(Exposure), 
    Number.Observations = length(Exposure))

We can show this on a plot as well:

ggplot(Power.summary, aes(x = Power, y = totalExposure, fill = Power)) + geom_bar(stat = "identity") + 
    ylab("Exposure in years") + geom_text(stat = "identity", aes(label = round(totalExposure, 
    0), color = Power), vjust = -0.5) + guides(fill = FALSE, color = FALSE)

Let us now look at the observed claim frequency in each level

Power.summary = ddply(dataset, .(Power), summarize, totalExposure = sum(Exposure), 
    Number.Observations = length(Exposure), Number.Claims = sum(ClaimNb), Obs.Claim.Frequency = sum(ClaimNb)/sum(Exposure))
Power.summary

Power	totalExposure	Number.Observations	Number.Claims	Obs.Claim.Frequency
d	18711.53	33720	1148	0.06135
e	22157.35	38329	1631	0.07361
f	27823.54	47709	1998	0.07181
g	25678.47	45390	1672	0.06511
h	6956.80	13346	505	0.07259
i	4680.11	8793	369	0.07884
j	4587.50	8962	357	0.07782
k	2247.44	4659	188	0.08365
l	1046.09	2255	78	0.07456
m	475.74	905	37	0.07777
n	317.64	624	31	0.09760
o	333.32	740	26	0.07800

We can compute the ratio to the portfolio claim frequency.

portfolio.cf = sum(dataset$ClaimNb)/sum(dataset$Exposure)
ggplot(Power.summary) + geom_bar(stat = "identity", aes(x = Power, y = Obs.Claim.Frequency, 
    fill = Power)) + geom_line(aes(x = as.numeric(Power), y = portfolio.cf), 
    color = "red") + guides(fill = FALSE)

CarAge

The vehicle age, in years. This is the first continuous variable that we encounter (although it only takes discrete values).

ggplot(dataset, aes(x=CarAge)) + geom_bar()  + xlab("Age of the Car")

Again, here, the exposures are not considered on the histogram. We can use ddply to correct this.

CarAge.summary = ddply(dataset, .(CarAge), summarize, totalExposure = sum(Exposure), 
    Number.Observations = length(Exposure))
CarAge.summary

CarAge	totalExposure	Number.Observations
0	4346.42	14947
1	9016.91	18863
2	8621.68	16246
3	7902.55	14337
4	7438.18	12851
5	7206.40	11889
6	6920.09	11245
7	6538.41	10516
8	6636.01	10624
9	6319.48	10268
10	6931.74	12168
11	6051.96	9727
12	5819.12	9544
13	5622.09	9169
14	4938.09	8244
15	4349.96	7674
16	3099.81	5156
17	2375.68	3950
18	1713.78	2822
19	1175.23	1931
20	732.41	1231
21	470.92	781
22	316.97	505
23	207.92	328
24	145.56	219
25	118.17	197

Then, we can plot the data onto a barplot, as before.

ggplot(CarAge.summary, aes(x = CarAge, y = totalExposure)) + geom_bar(stat = "identity") + 
    ylab("Exposure in years")

We can see a large difference, specially for new cars, which makes sense ! Indeed, let us look at the Exposure for new vehicles, using a boxplot for instance.

ggplot(dataset[dataset$CarAge == 0, ], aes(x = "Exposure", y = Exposure)) + 
    geom_boxplot() + ggtitle("Exposure of new cars")

Let us now also compute the claim frequency by age of car,

CarAge.summary = ddply(dataset, .(CarAge), summarize, totalExposure = sum(Exposure), 
    Number.Observations = length(Exposure), Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
CarAge.summary

CarAge	totalExposure	Number.Observations	Number.Claims	Obs.Claim.Freq
0	4346.42	14947	285	0.06557
1	9016.91	18863	653	0.07242
2	8621.68	16246	591	0.06855
3	7902.55	14337	551	0.06972
4	7438.18	12851	528	0.07099
5	7206.40	11889	501	0.06952
6	6920.09	11245	509	0.07355
7	6538.41	10516	502	0.07678
8	6636.01	10624	479	0.07218
9	6319.48	10268	487	0.07706
10	6931.74	12168	527	0.07603
11	6051.96	9727	441	0.07287
12	5819.12	9544	449	0.07716
13	5622.09	9169	369	0.06563
14	4938.09	8244	330	0.06683
15	4349.96	7674	273	0.06276
16	3099.81	5156	197	0.06355
17	2375.68	3950	140	0.05893
18	1713.78	2822	88	0.05135
19	1175.23	1931	48	0.04084
20	732.41	1231	39	0.05325
21	470.92	781	20	0.04247
22	316.97	505	11	0.03470
23	207.92	328	6	0.02886
24	145.56	219	9	0.06183
25	118.17	197	7	0.05924

and plot it!

ggplot(CarAge.summary, aes(x = CarAge, y = Obs.Claim.Freq)) + geom_point() + 
    ylab("Observed Claim Frequency") + xlab("Age of the Car") + ylim(c(0, 0.08))

DriverAge

Similarly to the Age of the Car, we can visualize the Age of the Drivers.

DriverAge.summary = ddply(dataset, .(DriverAge), summarize, totalExposure = sum(Exposure), Number.Observations = length(Exposure), Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
head(DriverAge.summary,9)

DriverAge	totalExposure	Number.Observations	Number.Claims	Obs.Claim.Freq
18	78.37	257	23	0.29347
19	314.15	811	90	0.28648
20	492.48	1213	112	0.22742
21	637.28	1528	105	0.16476
22	794.64	1822	122	0.15353
23	923.73	2074	124	0.13424
24	1073.46	2434	111	0.10340
25	1234.65	2765	114	0.09233
26	1476.38	3240	157	0.10634

We can show the Exposures by Age of the Driver

ggplot(DriverAge.summary, aes(x = DriverAge, y = totalExposure)) + geom_bar(stat = "identity", 
    width = 0.8) + ylab("Exposure in years") + xlab("Age of the Driver")

and the observed claim frequency by Age.

ggplot(DriverAge.summary, aes(x = DriverAge, y = Obs.Claim.Freq)) + geom_point() + 
    ylab("Observed Claim Frequency") + xlab("Age of the Driver")

Brand

The variable Brand is a categorized variable, related to the brand of the car. We can see the different levels of a factor by using the function level in R:

levels(dataset$Brand)

## [1] "Fiat"                              
## [2] "Japanese (except Nissan) or Korean"
## [3] "Mercedes, Chrysler or BMW"         
## [4] "Opel, General Motors or Ford"      
## [5] "other"                             
## [6] "Renault, Nissan or Citroen"        
## [7] "Volkswagen, Audi, Skoda or Seat"

Brand.summary = ddply(dataset, .(Brand), summarize, totalExposure = sum(Exposure), 
    Number.Observations = length(Exposure), Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
Brand.summary

Brand	totalExposure	Number.Observations	Number.Claims	Obs.Claim.Freq
Fiat	4728.26	8341	357	0.07550
Japanese (except Nissan) or Korean	15499.62	39413	993	0.06407
Mercedes, Chrysler or BMW	5254.51	9651	416	0.07917
Opel, General Motors or Ford	10812.52	18638	847	0.07834
other	2905.09	4947	232	0.07986
Renault, Nissan or Citroen	66756.93	108267	4446	0.06660
Volkswagen, Audi, Skoda or Seat	9058.60	16175	749	0.08268

require(ggplot2)
ggplot(Brand.summary, aes(x = reorder(Brand, totalExposure), y = totalExposure, 
    fill = Brand)) + geom_bar(stat = "identity") + coord_flip() + guides(fill = FALSE) + 
    xlab("") + ylab("Exposure in years")

Let us now look at the claim frequency by Brand of the car.

ggplot(Brand.summary, aes(x = reorder(Brand, Obs.Claim.Freq), y = Obs.Claim.Freq, 
    fill = Brand)) + geom_bar(stat = "identity") + coord_flip() + guides(fill = FALSE) + 
    ggtitle("Observed Claim Frequencies by Brand of the car") + xlab("") + ylab("Observed Claim Frequency")

Gas

The variable Gas is a categorized variable, related to the fuel of the car. We can see the different levels of a factor by using the function level in R:

levels(dataset$Gas)

## [1] "Diesel"  "Regular"

Gas.summary = ddply(dataset, .(Gas), summarize, totalExposure = sum(Exposure), 
    Number.Observations = length(Exposure), Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
ggplot(Gas.summary, aes(x = Gas, y = totalExposure, fill = Gas)) + geom_bar(stat = "identity") + 
    guides(fill = FALSE)

There seems to be a similar amount of Diesel and Regular gas vehicles in the portfolio. It is generally expected that Diesel have a higher claim frequency. Does this also hold on our dataset ?

ggplot(Gas.summary, aes(x = Gas, y = Obs.Claim.Freq, fill = Gas)) + geom_bar(stat = "identity") + 
    guides(fill = FALSE)

Region

The variable Region is a categorized variable, related to the region of the place of residence. We can see the different levels of a factor by using the function level in R:

levels(dataset$Region)

##  [1] "R11" "R23" "R24" "R25" "R31" "R52" "R53" "R54" "R72" "R74"

What are the Exposures in each region ? What are the observed claim frequencies ?

Region.summary = ddply(dataset, .(Region), summarize, totalExposure = sum(Exposure), 
    Number.Observations = length(Exposure), Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
Region.summary

Region	totalExposure	Number.Observations	Number.Claims	Obs.Claim.Freq
R11	15024.48	34778	1266	0.08426
R23	1531.38	4275	94	0.06138
R24	50927.07	79767	3289	0.06458
R25	3274.99	5398	227	0.06931
R31	5676.75	13576	483	0.08508
R52	10871.82	19321	760	0.06991
R53	13889.48	21032	928	0.06681
R54	5573.92	9455	386	0.06925
R72	7033.85	15506	516	0.07336
R74	1211.77	2324	91	0.07510

Using the function twoord.plot we can easily show both the Exposures and the observed claim frequencies on the same plot.

twoord.plot(1:10, Region.summary$totalExposure, 1:10, Region.summary$Obs.Claim.Freq, 
    xlab = "Region", rylim = c(0, 0.1), type = c("bar", "p"), xticklab = Region.summary$Region, 
    ylab = "Exposure", rylab = "Observed Claim Frequency")

We can plot a map with the observed claim frequencies

# Download shapefile from http://www.diva-gis.org/gData Extract all the
# files from the zip files, in a directory called shapefiles in your working
# directory
area <- rgdal::readOGR("C:/Users/juerg/AktuarDataScience/Varia/2019_SSA/shapefiles/FRA_adm2.shp")  # From http://www.diva-gis.org/gData

## OGR data source with driver: ESRI Shapefile 
## Source: "C:\Users\juerg\AktuarDataScience\Varia\2019_SSA\shapefiles\FRA_adm2.shp", layer: "FRA_adm2"
## with 96 features
## It has 11 fields
## Integer64 fields read as strings:  ID_0 ID_1 ID_2

Region.summary$id = sapply(Region.summary$Region, substr, 2, 3)
area.points = fortify(area, region = "ID_2")  #Convert to data.frame
area.points = merge(area.points, Region.summary[, c("id", "totalExposure", "Obs.Claim.Freq")], 
    by.x = "id", by.y = "id", all.x = TRUE)
area.points = area.points[order(area.points$order), ]  #Has to be ordered correctly to plot.
ggplot(area.points, aes(long, lat, group = group)) + ggtitle("Observed Claim Frequencies") + 
    geom_polygon(aes(fill = area.points$Obs.Claim.Freq)) + scale_fill_gradient(low = "green", 
    high = "red", name = "Obs. Claim Freq.", limits = c(0.061, 0.085)) + xlab("Longitude") + 
    ylab("Latitude")

and the exposures (on a log-scale)…

ggplot(area.points, aes(long, lat, group = group)) + ggtitle("log Exposures in years") + 
    geom_polygon(aes(fill = log(area.points$totalExposure))) + scale_fill_gradient(low = "blue", 
    high = "red", name = "log Exposure") + xlab("Longitude") + ylab("Latitude")

Density

The Density represents here the density of the population at the place of residence. Let us take a look at the densities in the dataset.

summary(dataset$Density)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##       2      67     287    1982    1408   27000

ggplot(dataset, aes(Density)) + geom_histogram(bins = 200)

Here, contrary to the age of the driver, or the age of the car, the density has lots of different values

length(unique(dataset$Density))

We can compute this by using the command above, and we get 1267.

Density.summary = ddply(dataset, .(Density), summarize, totalExposure = sum(Exposure), 
    Number.Observations = length(Exposure), Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
head(Density.summary)

Density	totalExposure	Number.Observations	Number.Claims	Obs.Claim.Freq
2	9.07	13	1	0.11025
3	58.94	81	2	0.03393
4	18.22	29	1	0.05490
5	67.99	122	7	0.10295
6	131.76	207	8	0.06072
7	182.77	288	7	0.03830

We can plot the observed claim frequencies…

ggplot(Density.summary, aes(x = Density, y = Obs.Claim.Freq)) + geom_point()

… but realize it is impossible to see a trend. One way out is to categorize the variable. We will see later (GAM) that it is possible to estimate a smooth function, which avoid the arbitrary categorization.

We can categorize the variable using the function cut.

dataset$DensityCAT = cut(dataset$Density, breaks = quantile(dataset$Density, 
    probs = seq(from = 0, to = 1, by = 0.1)), include.lowest = TRUE)
table(dataset$DensityCAT)

## 
##             [2,28]            (28,51]            (51,91] 
##              20813              20571              20805 
##           (91,158]          (158,287]          (287,554] 
##              19986              20584              20508 
##     (554,1.16e+03] (1.16e+03,2.4e+03] (2.4e+03,4.35e+03] 
##              20571              20541              21290 
## (4.35e+03,2.7e+04] 
##              19763

levels(dataset$DensityCAT) <- LETTERS[1:10]

Then, we can apply the same strategy as above.

Density.summary = ddply(dataset, .(DensityCAT), summarize, totalExposure = sum(Exposure), 
    Number.Observations = length(Exposure), Number.Claims = sum(ClaimNb), Obs.Claim.Freq = sum(ClaimNb)/sum(Exposure))
Density.summary

DensityCAT	totalExposure	Number.Observations	Number.Claims	Obs.Claim.Freq
A	13499.50	20813	678	0.05022
B	12959.15	20571	738	0.05695
C	12695.77	20805	769	0.06057
D	12001.69	19986	754	0.06282
E	12166.47	20584	798	0.06559
F	11918.67	20508	845	0.07090
G	11065.29	20571	904	0.08170
H	10670.17	20541	897	0.08407
I	9707.02	21290	922	0.09498
J	8331.80	19763	735	0.08822

Using the function twoord.plot we can easily show both the Exposures and the observed claim frequencies on the same plot.

twoord.plot(1:10, Density.summary$totalExposure, 1:10, Density.summary$Obs.Claim.Freq, 
    xlab = "Density (categorized)", lylim = c(0, 15000), rylim = c(0, 0.15), 
    type = c("bar", "p"), xticklab = Density.summary$Density, ylab = "Exposure", 
    rylab = "Observed Claim Frequency", lytickpos = seq(0, 15000, 5000), rytickpos = seq(0, 
        0.15, 0.03))